Thin prism, dispersion and dispersive power
A thin prism deviates light by δ = (μ − 1)A. Glass has a larger μ for violet than for red, so white light spreads into a spectrum.
Derivation: deviation by a thin prism
- 1For a prism of small angle A, the angles i, e, r₁ and r₂ are all small. So sin θ ≈ θ, with θ in radians.
- 2Snell's law at the first face, sin i = μ sin r₁, becomes i = μr₁. At the second face, e = μr₂.
- 3Adding: i + e = μ(r₁ + r₂) = μA, since r₁ + r₂ = A.
- 4For any prism, i + e = A + δ. So A + δ = μA.
- 5★So, δ = (μ − 1)A: a thin prism deviates a ray by nearly the same angle, whatever the angle of incidence.
★ key step
Dispersion of white light
Sunlight is a mix of colours: violet, indigo, blue, green, yellow, orange and red (VIBGYOR). In vacuum and in air, all colours travel at the same speed, so they stay together. In glass, violet travels slowest and red fastest. Since μ = c/v, glass has the largest μ for violet and the smallest μ for red. By δ = (μ − 1)A, violet is deviated the most and red the least. So a prism spreads white light into a spectrum, with red nearest the apex and violet nearest the base. This splitting is dispersion, and glass is a dispersive medium. A parallel glass slab shows no spectrum, because the colours come out parallel to the incident beam and overlap again.
Deviation, angular dispersion and dispersive power
| Quantity | Formula | Meaning |
|---|---|---|
| Mean deviation | δy = (μy − 1)A | Deviation of yellow, the middle colour of the spectrum |
| Angular dispersion | δv − δr = (μv − μr)A | Angle between the violet and red rays; it grows with the prism angle A |
| Dispersive power | ω = (μv − μr)/(μy − 1) | Angular dispersion divided by mean deviation; A cancels, so it is the same for every prism of one material |
Worked example: dispersion by a thin prism
(a) A thin prism of angle 4° is made of glass with μ = 1.6. Find the deviation. (b) A thin crown glass prism of angle 6° has μv = 1.523, μr = 1.513 and μy = 1.517. Find δv, δr, the angular dispersion, the mean deviation and the dispersive power.
- (a) δ = (μ − 1)A = (1.6 − 1) × 4° = 0.6 × 4° = 2.4°.
- (b) δv = (1.523 − 1) × 6° = 0.523 × 6° = 3.138°.
- δr = (1.513 − 1) × 6° = 0.513 × 6° = 3.078°.
- Angular dispersion = δv − δr = 3.138° − 3.078° = 0.060°.
- Mean deviation δy = (1.517 − 1) × 6° = 0.517 × 6° = 3.102°.
- ω = (μv − μr)/(μy − 1) = 0.010 ÷ 0.517 ≈ 0.019. Check: 0.060° ÷ 3.102° ≈ 0.019.
(a) 2.4°; (b) δv ≈ 3.14°, δr ≈ 3.08°, angular dispersion 0.06°, δy ≈ 3.10°, ω ≈ 0.019
How to draw dispersion by a prism
- Draw a triangular prism with its apex at the top. Draw a narrow beam of white light falling on the left face; label it white light.
- Inside the prism, draw the beam starting to spread into a narrow fan.
- From the right face, draw the rays leaving. Draw red nearest the apex (least bent) and violet nearest the base (most bent).
- Extend the original direction of the white beam as a dotted line. Mark δr and δv between it and the red and violet rays.
- Draw a screen on the right showing the colours in order, red at the top and violet at the bottom, and label it spectrum (VIBGYOR).
The cut glass drops of a chandelier and a well-cut gem flash colours in sunlight, because each face acts as a small prism. Camera makers combine glasses of different dispersive power to remove coloured fringes from photos.
2 marks: “Define dispersive power. On what does it depend?” ω = (μv − μr)/(μy − 1), the ratio of angular dispersion to mean deviation. It depends only on the material of the prism, not on its angle.
2 marks: “A thin prism deviates red, yellow and violet light by 2.4°, 2.6° and 2.8°. Find the dispersive power of its glass.” ω = (δv − δr)/δy = (2.8° − 2.4°) ÷ 2.6° = 0.4 ÷ 2.6 ≈ 0.15.
1 marks: “Which colour of white light is deviated the least by a glass prism, and why?” Red, because glass has the smallest refractive index for red light (red travels fastest in glass).
Violet is very bent; red is rather straight. For a thin prism, δ = (μ − 1)A.